We give examples of specific quartics interesting in Noether-Lefschetz loci for K3 surfaces, and where they fit in the Betti classification.
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The first example illustrates Corollary 6.18.
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We illustrate Remark 6.19, considering a double quadric:
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Next, we illustrate Remark 6.20. The first example is that of the Vinberg most singular K3 surface. This is of type [331].
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The second example illustrating Remark 6.20 is that a general element of the Dwork pencil has type [000].
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The third example illustrating Remark 6.20 is that the K3 quartics $S_{t}\subset \mathbb{P}^{5}$ given by $$ x_{1}^{4}+\dots+x_{5}^{4}-t(x_{1}^{2}+\dots+x_{5}^{2})^{2}=x_{1}+\dots+x_{5}=0$$ for general $t$ are of type [000]. However, $S_{0}$ is of type [550].
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The source of this document is in /build/reproducible-path/macaulay2-1.25.06+ds/M2/Macaulay2/packages/QuaternaryQuartics/Section6Doc.m2:94:0.